Abstract: In this paper we introduce several test statistics of testing the null hypotheses of a random walk (with or without drift) against models that accommodate a smooth nonlinear shift in the level, the dynamic structure, and the trend. We derive analytical limiting distributions for all tests. Finite sample properties are examined. The performance of the tests is compared to that of the classical unit root tests by Dickey-Fuller and Phillips and Perron, and is found to be superior in terms of power.